Approximate formulas for the computation of turbulent boundary-layer momentum thicknesses in compressible flows

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Title:
Approximate formulas for the computation of turbulent boundary-layer momentum thicknesses in compressible flows
Alternate Title:
NACA wartime reports
Physical Description:
26, 8 p. : ill. ; 28 cm.
Language:
English
Creator:
Tetervin, Neal
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

Subjects

Subjects / Keywords:
Turbulent boundary layer   ( lcsh )
Compressibility   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: Approximate formulas for the computation of the momentum thicknesses of turbulent boundary layers on two-dimensional bodies, on bodies of revolution at zero angle of attack, and on the inner surfaces of round channels all in compressible flow are given in the form of integrals that can be conveniently computed. The formulas involve the assumptions that the momentum thickness may be computed by use of a boundary-layer velocity profile which is fixed and that skin-friction formulas for flat plates may be used in the computation of boundary-layer thicknesses in flow with pressure gradients. The effect of density changes on the ration of the displacement thickness to the momentum thickness of the boundary layer is taken into account. Use is made of the experimental finding that the skin-friction coefficient for turbulent flow is independent of Mach number. The computations indicated that the effect of density changes on the momentum thickness in flows with pressure gradients is small for subsonic flows.
Statement of Responsibility:
by Neal Tetervin.
General Note:
"Report no. L-119."
General Note:
"Originally issued March 1946 as Advance Confidential Report L6A22."
General Note:
"NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

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University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003636165
oclc - 71777608
sobekcm - AA00006247_00001
System ID:
AA00006247:00001

Full Text

ACR No. L6A22


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS





WAlTIME II REPORT
ORIGINALLY ISSUED
March 1946 as
Advance Confidential Report L6A22

APPROXIMATE FORMULAS FOR THE COMPUTATION OF
TURBULiNT BOUNDARY-LAYER MOMETUM
THICKNESSES IN COMPRESSIBLE FLOWS
By Neal Tetervin

Langley Memorial Aeronautical Laboratory
Langley Field, Va.









'*, : .i- : ; .. .

WASHINGTON

NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of
advance research results to an authorized group requiring them for the war effort. They were pre-
viously held under a security status but are now unclassified. Some of these reports were not tech-
nically edited. All have been reproduced without change in order to expedite general distribution.


L 119


DOCUMENTS DEPARTMENT


WAt l-)(1





































Digitized by the Internet Archive
in 2011 with funding from
University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation


http://www.archive.org/details/approximateformu001ang




- 1 -14 -i -


T.'A .':. Ho. L6A22

:. -IC'ITLI ADVISORY CO'.nITTEO FOR AERONAUTICS


-r '. '_-,' .'' ,,F'iZE;'TIm .'r F;I'' 9?, CT


NPrROXIr!ATE7 FOR~1T AS FOR THE COMPUT TIOI OPF

7TR E'TIUiT EPOn'TDARY-L AYER TMOMBTUM

THICv.ICSSES IIH 00 ,PRESS IBE FLOcVS

By 'Ieal Tetervin


SUMMA L RY


Apprczimate formulas for the com-ut'tion cf the
moment'ru thicknesses of turbulent boundary layers on
two-dimensional bodies, on bodies of revolution at zero
angle of attack, and on the inner surfaces of round
channels all in compressible flow are given in the form
of irtec-r-ls that c0-n be conveniently computed. The
formulas involve the assumt"itions t.at the mc.mentWun
thickness may be coi~puted b u.se of a bcundary-layer
velocity profile which is fixed: .and that skin-friction
formulas fcr flat plates nay be used in the computation
of 'oundary-layer thicknesses in flow with pressure
gradients. The effect of density changes on the ratio
of the d-isolacement thickness to the momentinm thickness
of the -boundary lIver is t-ken intc account. Use is
made of the experimental finding that the skin-friction
coefficient for turbulent flow is independent of i'ach
number. The comoutstions indicate that the effect of
density changes on the momentum thickness in flows with
pressure rra.'ients is smRall for subsonic flovs.


TI.TRODUCTIO'7


A nLLmber of methods are available for the computation
of boundary-layer momentum thicknesses for inccmoressible
flow. ThE increasing importance of flows at Mach numbers
approaching and exceeding 1 has emphasized the need cf
formulas that r.oul:l ?-alke possible the comparatively rapid
computation of boundary-layer momentum thicknesses for
corpressible flows. The purpose of the present work is
therefore to provide approximate formulas for the compu-
tation of boundary-layer-thickness parameters for








2 CONFIDENTIAL NACA ACR No. L6A22


compressible flows. The present work furnishes no new
information concerning the boundary-layer shape, skin-
friction coefficient, position of the transition point,
or likelihood of boundary-layer separation.

Approximate formulas for the computation of the
momentum thicknesses of turbulent boundary layers on
two-dimensional bodies, on bodies of revolution at zero
angle of attack, and on the inner surfaces of round
channels all in compressible flow are given in the form
of integrals that can be conveniently computed. The
approximate formulas contain the assumptions that the
momentum thickness may be computed by use of a boundary-
layer velocity profile which is fixed during the inte-
gration and that skin-friction formulas for flat plates
may be used in the computation of boundary-layer momentum
thicknesses for flow with pressure gradients. The
formulas are applicable to all unseparated, turbulent
boundary layers and in special cases to laminar boundary
layers. The numerical values of the ratio of the dis-
placement thickness to the momentum thickness, a ratio
that appears in the momentum equation and that is capable
of specifying approximately the velocity distribution
through the turbulent boundary layer in incompressible
flow, are corrected for density changes in the boundary
layer by use of low-speed velocity distributions. Use is
made of the.experimental finding that the skin-friction
coefficient in turbulent flow is independent of Mach
number.

The problem of computing the boundary-layer momentum
thickness for compressible flow has been treated by Young
and Winterbottom (reference 1), who integrated the
boundary-layer momentum equation for laminar flow by using
the skin-friction relation from the Pohlhausen theory
(reference 2, p. 109), fixing the velocity profile, and
correcting the density through the boundary layer for the
effects of compressibility. For the turbulent boundary
layer, the momentum equation was integrated by a step-by-
step process in which a fixed velocity profile was used
and the effect of density changes through the boundary
layer on the ratio of the displacement thickness to the
momentum thickness was ignored.

The problem of commuting the momentum thickness over
a body of revolution for incompressible flow has been
treated by Young (reference 3), who computed the momentum
thickness of the laminar boundary layer by a step-by-step


CO7IVDE.ITIAL









NACA ACr :*-''. L6A22


con:,'-it ; :in in ',I- I-ih an -e:-tens":on of the Po l, use-n r.ethod
wa. used. 'I'e thi. boIin"-'a'.ry layer .'*' -Co:r",ited b- .9. ?te<- -:- te, praoeer t in
wh-ch a fi:e'J -el-c.'-- .it ; -fil.- in ths ior:rentur. equat ,ion
f3;i = t,..',j, cf' :e'..' l: ti.:.-.- ,:'er.- ".1 ,.
for a :d. off eluticn smrz i-o.

In order:. to sj ~tsntiate the 9.. 2:r- Ci L th- .t F in-
friction fo.ru:.ls a ,s r tr rl..i ent fl]o:. aloni flat pr -tes
rr, a he usE:. in t'r.e 'n ,,tf:-'t ion f n m.:.nt ur' thicknesses
for flo,_, wvit-h ires-c:uLre rai ients, r.liferLnncs : to o are
cite' In thj-se r f i r nc'es, I.c.C S r re.Si: -nrt ',-t':,een calcu.-
latce an,1 ey:,:' ri-re:-',sl rer--' ts 1 'A,:- :ener.s]1.-r ,obtain-ed
Slt!ou.,lh faii;-rly '" ajvF-rs: .reC E s.r? c;9 j.ts were
present in man1' 3f the cRes.

The as s .. -t :.n that thc .iorrentrPi' thic'e ncl ess n Z I be
coir:'lt:. :ei to a c ].C' 9 : :,': 'i: iatOto, 07n f1.r:in1 the velo,'itr
orofile 'o.i. inj; inter tiocn is su:stsnt iate t c:, the v,'rk
in refer.rcnc::'s s sid i, and by rlT; t in rrCc [er-enc,: 6, which h
contains E. ce,,:maro._ bc. 1 'een the co:m 'itedc: arn e '-eri-
mental : .'; iec if t mrrm=nt'r thick' s over the entire
chcrd.

That the sliin-frict ior cc cf ric ien:t for tirb!ul.nt flow
is inde.'-endc.-nt cf !'ach number is established b th;. 'Aork
in r-eferenc s 7 n F'rc. 'el i'eference 7 rre-; ents
e.-:pe rime.nt al d t t f',or turbuil -rnt "icv'. -a Eies ,hi,. show
th' t tne vel .city profiles for lisubs: 'n.LC C O.Fri Zsi I e flow
and tie skin-fri':c ion cor fficienti for nbs.:iJc and 'surer-
so.liiC Corr.' ressir:le flow '.i not dififr a cticea-aly from
those for ineir. sle flo Th-ecJdoi en and Remier
referencee '} b.; xer.lriment ifr .lth ro'.tttinC 'is -.:,., a'iov;ed
that the kin-fr Ltion coeff icie-nt for turl:-ulent boundary
layers is irZndeFp ndent of M!'~i.:h nurmier. ITeena.r. and Ireaman
referencee ) after *:erforming e':.-riments \ ith pipes,
reached ccnicl1sion: that c id not cor.trailct those cf
references ,7 -lnd i.





a con-t -nt i- eqiati:'n relating Pc to \ and E-

b slo .e of' ;c-locity I1st1i tirit on

ed ddrao coefficient '.er unit span


CCITTDTPF 71~IrTT


C O'F' ID' TI AL








ITn -A AC1 NTo. L6A22


co velocity of sound in free stream

C-, specific heat at constant pressure, foot-pounds
per pound-mass per degree

TF ratio of displacement thickness to momentum
thIickness (-/e )

K constant

k constant in skin-friction formula

L length of airfoil, body of revolution, or round
channel; measured along chord or axis of
revolution

Mo free-stream Mach number

m exponent in formula for boundary-layer velocity
distribution

m' particular value of m

n exponent in skin-friction formula

p static pressure

R gas constant

RL Reynolds number (UoL/Do)

Rx Reynolds number based on length of plate (TJoX/Uo)

Re Reynolds number based on momentum thickness (u'0/-J)

r radial distance of point from axis of body of
revolution or round channel

rt radius of body of revolution or round channel

rtax maximum radius of body of revolution or round
channel

T .absolute temperature

To absolute temperature of free stream

T6 absolute temperature at edge of boundary layer

T T'DT-.:T' AT,


CONF IDENT IAL









7 ..IA AC:R :TI. L6A22 CTIFT~ITTAL 5


I veloc-ity parallel to x. at cuter e ,e of boundary


IU, frx' e-st re-= am ve locity

Uxy al of at station at ich value of is
obta "ned

T. v-.lue of TU -t x = D
vj
V va. lue of 1U chosen to make value of I,-- a
may imumi \/

U1 v.-lue of cT at a :0

u velocity inside boLundridr l-,-r .annd oar-ollel to
s'. .r- a c

w ;e:*.l"nent inr fcrr."ul a for viscosity

x distance :r-ure alng surface from forward
_.stl rn t ier. cint

x0 po:si ion on surface at be..:inin of intecgrtion

y distance m'-c ';r,'D- norm.; al to .x

a siol. e b. t'vree t'- c'ent to s'.ll9re cf bodyJ of
revolution or round channel ancd ax:is of
re vo1 ut on

F = rt- + i ccs a

~.1 val..ue of a st x-

v ratio o" sE:,ific heat at constant Di-res1ure to
s; c i fic hea t at cons c!n t volume

6 nominal tVhiclkneos cf bojunidary lDyer
r
5e' cc -1 1 t tIJ n

Srmomentui thickness P" i1 -u

91 value of 8 9t x,


CC. -TPP T IAL







PACA ACR Ho. L6A22


J(U/o)2

p. coefficient of viscosity
kjo free-stream viscosity
u cinematic viscosity at outer edge of boundary
layer

Vo cinematic viscosity in free stream

p density

Po free-stream density
p5 density at edge of boundary layer
To surface shearing stress

= + 2
(y 1)Mo2

S= rte cos a

i1 value of 9 at x0

d" =J (1 1y dy

: -f (1 PU e

Subscripts:

c compressible flow
i incompressible flow


ANALYSIS

Momentum equation for compressible flow about a two-
dimensriLcp-i tboo .- hr.- rc'nd r.y-i yr. ror oe ntmi equation
for two-dimensional compressible flow (reference 10, p.l.2)

CONFIDENT rI A


CONFIDENTIALITY









NACA ACR To. L6A22 COC~i'IlT -TIAL 7


s i v n, -henr the sttic-:,res ur'e -'ir-is t.ion across the
bo.in-' r- yr is nre li i e, a s



/ v'' d-, *T 7 l U .- c (1'


FrThom the cua3 t ,n 1r'",'t i- for c "'r." e s l it',isc
fio-" th-e rt. 1 O ion *..Yct,r., en th-e -;el,,.,, t; 'II, r e.. .re

for CIcr."e=: i T,? 9 S




,'.- :,
Theni b. ;1s-c of i tIhc ;cr'ati.. te tionr forP






and- the Ifi.L t ion f'or the ci l.c-i .. c -it ti ': ness
0 0
::- j*, (I- ')



equa.tion !-, 1 ca:-n ta ':. rI.ttn i, t'e for! ci-.:en in refer nce 1








".The .ri--C .e: ufse.d in th, .i.: ati,'r .f equ t on (2

were the conservation o.. mass ad aJevtor's law of moti-'n.
Equatio. (1) -s thereorc a -?uic.,ole t, both subsonic ,nd.
surers.nic 1. T. H + uti not, owe-. er to be
i', e r e






whe assnti-ons c.C the bof undcriass a.d ;rer ti oy "a; not be


CCFIT-i'TI 41L








SHA CA ACR io. L6A22


;: -rientutr equation for comnressible flow about a body
of revolution.- The boundary-layer omrentum equation for
comnressible flow about a body of revolution (reference 10,
p. 13) can be written, when the static-pressure variation
across the boundary layer is negligible, as


pu2r dy U -- j
6x Jo


pur dy = Tort -r dy
O I r


r = rt + y cos a


where rt and cos a are dependent on x only
equation ( ) ""7 be rewritten as


Spu2rt dy U -
\+ x JO 6xJo


+^ J p u2fcsyay 'Lr J


(fig. 1),


purt dy



pu (cos a) ydy =


x JO rt dy C I cos ay dy rtTo


If 9 and P are defined by the relations


rt9p6U2 pu(U u)rt dy
and

C(cos a) pgU2 = o pu (U u) (cosa).y dy

and the equation of motion for compressible, inviscid flow


6 = PIT6U
6x 6x


CONFIDENTIAL


ix J

Since


COl,. ID:,: TIAL








"-.'A _.-'r: D. L6A22


CO iTfC 1lITIAL


is u-ed, th._en tlihe morne1.nt.im equation bec,:mprs


pUrt dy '
C L 0


purt d -


(rtrspu2)


pOTT ico; aO.'- d- /
j


-1
put beso) dy .
"i &J


- -- cs O.p,. I = rtro
Lt- J
L;


and '-": .-r; be dief re:i -; the r"el=< i:ons

t
rt :.'p t= r -p1r' pi)rt d.n


and


('e" C. .)p6TT


S (
- !0 !p;, u) (ec ) ,: c.'y


so that th-e t-rl.eiL rr: :i':ati-'n be, ''.es


+ 2 ccs 2 +


i P-
P;F ici


1 ---,rP 0 1- K Q+
-r -^^ ^-c + 2) mo=

StT 0,
(rte + ':os c) =
T -T


In cr-.-''ier t .-r i t: r- "- : .c th e .or entum
equaticrn f: :- ct .:.re si ie "C.:-'r a'..i t t. cd :.f te.oir icr.,
equation ( '. hu-.u a be '. r-itter in the s. e f,-rrn as that
for. t'"'o.- i,",'ersi?. .-i.c ca:,r. s.:;: 1 1T ._-l eou.ati:nr (i2)). The
a-r'i.,-xr i.st,' tl'i i tt-er'fcre !:c-ce li-st .-r fl'.: rn the body
of r e.Toi. ., i,-,,n


trtr(F + 2) +c s C) + -r


Cn? 'TTrID.'TTIL


- T


T 7


Then 6'


- (rt
dx.


?i(ti


(4)


= (:, + 2)(rte + C cos a)>








10 CONFIDENTIAL ITACA ;A^ ITo. L6A22


or

2 cos a~ 2
1 + 2i)
rte \iic +2 /
= K
1 cos a
1 C
rte

In order to determine the magnitude of the ratio
+ 2
Sthe assumption is made that the velocity distri-
Hc +2'
butions through the boundary layer may be approximated by
power curves of the type

u /y 1/m


Vt'hen the flow is incompressible, the definitions of *,
Q, and Hi may be used to obtain

S_1 +m


and
2 +m
m
Therefore

i + 2
_i 3m + 1
Hi + 2 3m + 2

for incomnressible flow.

From

/ F\y/y dy = f1 F 1 dy
do 2 Jo

it can be shown that


-M m=2m H
Sm=m m=2m'


CC' TDT17 "IAL









I-.'A .'?? 1o. L6A22 CCiFF D.' TIAL 11


flar c-'.i:.rezs 'bl as ."ell as f or inc?~: i rIes i ible flow.
- -js., cf





and ft:':re 2. v-hi,-h is .isc'.-sed later in connection iTAd

S+ 2
the ef t of c .neszziClit c -; e ra-le
K, + 2
Caii be ev T.-.-ate fr cn. e _sble f'.low. i r all CAse3 of
+ 2
uinse-araited f1cv.. th'e vle f the r.tin ---iffers
H + 2
frcii unit-. b less }- n 1C .'c:-c ., f 1or r iost cases, bI'
S a c,,h3 ,
rou hl7 1.' 'erc-nt. Thus, e-en .:en is not a

'- fr,- .- .;i- Is not_
small frPcti'in, t.? r3:rc-:i.mwtica :h?.t 7 = 1 is not
far fri trius. .1er, in tadd.'ition, it l noted that



t tte a .ro i-

and. i Lt-ler-.e -',-,re s3all over mcst of -he boed:, a Fe apro i-:i-
mraticn th.t = 1 is scorv'i:'- le. If K = 1 is used
in eq Lu tion (1), t :e r.ies

Ir~. + n cos c + + 2 rt + Pos a + c to + cos =
_x_ "_ x\ +" S,_Ox


rtTo



or u titu tlicn of .- r' + ccs a

;- A. + 2 0- 1 c r-tT,__,
+ + -- 5)
o.:: U %:" pg o -, p,,_U-


This *'.-atio'f for' ir-co:nr.ess icle .f'iLo.: has been given in
re :e ren-e ..,
r, e e n, -2
Ter principles .u.s. in ncLe erivatior. of equation (5)
;vere t-he conservation of mass and .:ewton's law of motion,
The apDr-oxim.ation was made that-. for flo.: on the body K = 1.


20 I-IDE ZlTLAL








NACA ACR jNo. L6A22


momentumm equation for compressible flow over inner
surface of a round channel.- 7fhen the flow in a round
channel is such that the flow throughout the boundary
layer is approximately parallel to the wall and a region
exists outside the boundary layer in which viscosity has
no effect, the boundary-layer momentum equation for this
flow is the same as equation (3). In the momentum equa-
tion for flow over the inner surface of a round channel,
the pressure change across the boundary layer is assumed
to be negligible. All of the symbols in the momentum
equation for compressible flow over the inner surface of
a round channel have their meanings unchanged except for
the distance y, which is positive when measured inward
from the wall so that

r = rt y cos a

The derivation of the momentum equation for flow over
the inner surface of a round channel follows the same pro-
cedure and involves the same assumptions as the derivation
of equation (5). The derived equation is
_] + 2 6U 1 6p_ C rtTo

6x U 6x Pg bx p6

where

S= rt O cos a

Effect of compressibility on Hc.- Since the ratio
of the displacement thickness to the momentum thickness
H, occurs in the momentum equation, the effect of com-
pressibility upon this ratio should be considered. The
equation for Hc is

-c


CONFIDENTIAL


CONFIDENTIAL









IACA ... 0. L6A22 T.ID'TT IAL 1


S i t-- ari aton u t-he .r.unr-iar- laer-n
S.r t -l.-. '-rm the ve loci;ty i t'"rition th rou-~.h the
':ou'a"'- I'J -er by r;str: acting the treatment to the case
in i.lich ".ere is Irn h? at flo- t'-Luch th su-'face and
the ef. -ctive Fran.-ict nur.be- s eqv.l uc nity. The
eq .iat ion

c -i' + cc-L*strt
t-

is u--e- r 4. ie-a.li to fl .,.' in he :..cui '-ur'y, la'rer. ,,r,_e:"L
t _e i ...itQ,"-..1 res i:-i;vt .c:n *C. .at the st-at c- n.iesz'.a
,.'a"r_.:t r. :.n '- ; r ;h the bo, xn:ia:r' l_:-er.. iz ne*-li.:ihle, the
I--- j 9. e S

p p


rc li '-
0 f'













chrcn induced tL
'1 -,







Ii i-a- 1- <- -
:c: r 2 ").T '




,I5 1



i + Y J3 i o -
0 b

"-- i 1



3. L \ 3 I ,. -
L:



T.AI








TACA ACR No. L6A22


1 + X-! Mo2 ,

r 1 2
1 + 2 bM
*141


(TJ\2~" s


where


2
- =1 +o
Yr 1) 2o2


then


-2


By application
of 6-/65, then


of the same procedure to the definition


2


U,


(8)


Large values of \2 mean s,iall values of Mo.

The assur.tion was made that

u /y\l/m


and values were chosen for X2; 6'/5 and 8/6 were then
calculated for a range of X between 1.5 and 14.0
with m 5, 54, 5, and 7. The curves of 1/Ec are given
in figure 2. Since

2 +m
fi -
m


CCOP i '-"'TT T L


Let


(7)


CONFIDENT IAL









IAC A -.Cf No. L6A22 CC,'PrIDrT; TI Ul. 15


an:i. ir.ce I- :pe ? r in tl'e mn':; nt.rI e.qusatioin, the curves
of figure 2 s-re designTiste. by,r valus of l- rather than rn.
Pc,. er c ,rv.-s sre usec' *rc-ri-ly for c. orvenl en1 ce6 n corriA'.ting
th -: e ff c.t .f c:,- :. :-r i b3 i] t H.

The -i.i of Fc for the Pi~--ius f'iet-:late .irof'il
for l msn; r b o. '. ,'s-y 1 :.- rs i f r-nce 2, .*'- b een
cor .ute.-; for 'sriou. '. -i ues of \ and s Prandrtl n.irrumber
of iniit-; t :v u ,- of equ atiins ( 7' d (ii). Ti'he cornm utations
wer-- re. *.t tii th e. "el the r .el ..t" :i.St iliticn for lamin:sar
flov- o''er S fla.t o ~ 3.:t = (refern 11). 'he
HCe ''.'a ;:. tte-, sa air nst \, the- results of both c,. minput ai-
tions w'-r ;-r s r ctical. 1 icer.tic-]. ftr C il .i eq s of .
On'1 th resu. jt? :bti;,n.-: fo'r the 31 2rsius profile are
there ore pr-.rsente : ;i fiure r.

Eq'. ti onl ( c.i '-F s-ov that 1 tlhozh the -.elci-it
distributtion t.i-roi- the be wilEry1 ler- 1 s med to be
inde-end.ent of s.: citln .lc nrfc th s.rfce, He may var
with sui fs, .c sit:.:'- .:eO'-.s.? cf it '- : ': .n-der e on .
For inte-ra tior. o-.f the rmr. cnturr uj ti- n, the de :enridence
of B- .-n \ r',--r be tsL.er, into acou'.- bv a -.ro:i.r t ingr
the curi-.u"s o f f'fi.rres 2 sri j 7 i.i. tio.n


c + H-
S- 1

1+
= ,. ] + H i (o'


T I h


The vl'es s: f a chob.sen to fit t-,e c rv.es of fisc ur-cs 2
and 5 .'ith s efficient accu'jrs-;y c ver r,,ost of t-e srane of
S ar Ie lGotte; a -inst Hi- In fiLire '+.

Inte.rat-o.:-: f mor- ntu e .. ic n f r t .,--d nen sional
flcw.- P beforee e.'u:of-ti.-2n (2) .'ar ur i:.terst 'd, (He+ 2),
1 'o Tr i
S, a,' oulj e r'- e : fi.r.ctio s of the
Pa ox p- 2C

number, sn- the momentum h. Th term He + 2 is

replaceFd by its ec- iv-'lenr .- + (I-- + 2_ Br se



0 .
C 'I I -'! T P` T TM








NACA ACR No. L6,22


of the gas law

PF P
RTg

the equation of motion

6p \u
6x P 6x

the Bernoulli equation for compressible flow
TT2 Uo2
cPT6 + = coTo +

and the equation for the velocity of sound

Co2 = (y 1) cpTo

the term 4 P can be written as
Pn 6x


1 P6P _
P6 6x


Uo- I

- T o


To
The local skin-friction coefficient
P6U2
as a power function of the local Reynolds n
on the boundary-layer momentum thickness by

k k -
P5o k Rn
psU2 Rn9 U


(10)



is expressed

umber based


(11)


By analogy with the work of reference 11, the viscosity
and density used to calculate R9 are those at the outer
edge of the boundary layer. The viscosity at the outer
edge of the boundary.layer is assumed to be given by


Ro \To/
P = /61w
110 TO


(12)


where w = 0.768 for air (reference 11).


CO0tI T..nTIAL


COI.TT FI DEIT T iAL


v v--








HACA -C.- '.. L6_-22 CONFIDENTIAL 17

The densit-.-: t t-e outer e'o-e of the boundary layer is



ePzJ ?I \i- ri
(l1,
P6 3' 1

in :'-iLci- the flc. ov .t'ide the boun; dalr y la'.r is ass'.u-.ed
to be s i b.ti,?. '.itions (12; pid ( .1 ) are used to.
give tr ere ir v s costt. in c.:-ustorn ( i1 as
funct -icn :'r the velocity -t-iL tio '101. ver ti-e bc- sn nd
the free-stresai Iv- .' r-,"Qr, -r. Z.-ouatiur (2 I t';he then be
vwritteFn r s
,- ,. / \ -
9 a \u :,'' U ,L .. :, k 1:,n'
a (, 1; / I -
6x 'I- _v A) ,-i r L F oIl
T j



This equtlicn is d ifferef't~l equation of the Bernourlli
type. i" 2n it is m*cde line -'r n n1d inte 2ra te : ':- staE ndard
methccs, t: e result is F

'- \___,'L. P _L _
.L (-f + ) -L






(/ ) 1/ -n' I

"i -
i 1+I 1
\- TJ -
-J I-

Th i s /c- I s -1 P+n
1" :a 1 k i 1
\L/ n \c. C

ix '*1' ( I+n+j + l+
17" 1 n


CO'TID:FTT IAL








ITIAC A ACR HOi. LoA22


where 1,/L and UI/Uo are the values of e/L and U/Uo
at = -. The value of is always greater than 1.
L L
By use of the Bernoulli equation for compressible flow,

(-o can be shown to be always less than -hen Mo
approaches zero, equation (14) becomes equation (1) of
reference 6, that is, the integrated momentum equation for
incoumpressible flow.

Tr-t. ,-r 9.t i. of -'t.,..rntr rr, r f':r flo '1. over a
body ,if 1.-';,-'. C' ho. .' n. "-,, a_ ; .- : *. .ii s Cj :lCc _-U"Fi Te- are
the same as those for the two-dimensional case. The
equations for c, .T-, and -6p are also the same
o P6 P6 ex
as for the two-dimensional case. The expression for the
skin friction, however, involves the apprcximation that
6 can be replaced by --. This approximation is
rt
equivalent to neglecting the term cos a in the
equation

0 Cos a
rt rt
Q 5
Since the value of t cos a is always less than -9e,
it follows that the approximation is justified only in
6
cases in which -- is a small fraction of unity. In
rt
regions near the tail, therefore, the accuracy of the
approximation for the skin friction may be expected to
decrease. The momentum equation for flow over a body of
revolution (equation (5)) indicates, however, that the
contribution of the skin friction to the boundary-layer
thickness becomes less important as the tail is approached.
The approximation that 0 can be replaced by is
rtTrt
therefore allowable and the term t--0 in equation (5)
P6U-
may be written
n+1 n
rtTo kt n
p(15)
pgTU2 pnun


C r T17 T-'r 7 mT T.,L


CO~T R'IENTIAL







FACA ACR io. L6 .'--1


1 aps rtTo
VWhen the terms Hc, P 6x and are replaced by
PO 6x a PgU2
equations (9), (10), and (15), then equation (5) becomes
TUU -
~U 2 D o
c+ a t. Hi + 2 To : =
Sx Ft ',U r 2
UG \t Uo j
*^ ^


rtl nkn


This equation is a di ffe-.ern:t ia e iueitn of the Bernoulli
type. '."hen it is macne linear srnr inttz:rated b-r standard
method s, the result is
i \ fa


i ,\


, -~ ,


. .


. L


2' 1

xL I
J


k(1+n) / L \l+n
R- k)n tmax

\ Y-1


rt '1+n.


+i)(i+n +'+ilr ]


-, l+n)
T /I i
Uo


'i (1+ri)
J,
i-71-t1raz


L
wh re -- ard
2
tma x

U/Uc .t --
L L


i+21 1 2,- 1+n 1

V- 0 -1
Sa( i+ni+n
\, '-l


IT/,/o re the values of 2 and
t max


CC';r, I'D 7T TA.J


CON IDENTICAL


.i
'T'1L 'j)'"







NACA ACR Ho. L6A22


Integration of momentum equation for flow over ifrer
surface of a round chennel.- The equation resulting from
the integrationn of equation (6), when the same procedure
and assumptions made to integrate equation (5) are used,
may be obtained from equation (16) by replacing p by
/. Tn the equation for 1/, the quantities denoting
reference conditions, Uo, rtmax, and M,, are the condi-
tions at a convenient reference section of the channel
and the length L is any convenient length.

The -definite integrals occurring in equations (14)
and (16) and the equation for i may be evaluated by
either an analytical or a rraohical method, whichever
is more convenient.


DISCUSSION


Before equation (14), equation (16), or the integrated
equation containing can be used, it is necessary to
know the velocity distribution over the surface, the con-
stants in the skin-friction formula, and the value to
choose for Hi. The velocity distribution rust be that
for the Miach number and Reynolds number for which the com-
putation is being made.

Skin-friction formuil.s.- For the turbulent boundary
layer, a pow-er function of. Rg is used for the skin-
friction formula. Because references 7 and 8 indicate no
noticeable effect of MKach number on skin friction, a skin-
friction formula for low speeds may be chosen and
approxim ated by

TO k
psU2 R n

as outlined in reference 6. One formula of the required
form is that of Falkner (reference 12)


To 0.006535
p5JU2 Re1/6

If the skin-friction data available are given in the form
of cd ~ Rx, they may be converted to Re by use
pgU


CO!T. IDE TIA I,








iAC. kCR 0 0o. L6A22 CONFIDENTIAL 21


c.f che rel"ti "u


n -
Tr 1ic -d
\


r alr:LnE r bcndar. v_ la'yjr.- ." thc'u h e. .,tin.ri ('L ,
equating ic. ,nd: the tqu .ion for h'.e been .derived.
for turbulent ';1oun'ci'ry L''-ers, these eqiati cns mey b used
for the a-prox.T'i1e c-',O- s ti .r1 of ar in
lanil nar tcUndr ': ly e-r .- t .hS, Frr r-i.A e choice of a
value for Hi an. ,..f a s-in-i rict' on formula. Althcurh
by use of the sl:in-fr ict.rion relaticn f.r the S i&..isi flat-
plate pr-ofile the sffect cf -rE. sur-e .adient o.n the skin-
fricticn coefficien- is nrlectrd, the -rror introd uced
is small so. lon- s Is the .vera-e oressur-e gra-,'ient over the
extent af tha iamin-r bo.; dsr-,- larver is Erriall. Thle skin-
friction ieiat ion trt n is

TO 0.220

p T T2 R,
6&

where 1: = 0'.2 a-d n = 1. Hlthclu. h sma T! l dec-rease
occisrs in k 1 s Ltie a3h ni-,ber incre s i refercnce 11),
the value cf 1 fcr inionr'. ssitle fl.,' (0.22C') may be
used in -' of t e a, :.r,'-c.- i cTio l irs .r ,naile coic rntr,,
the sk .n fri:ti:n.

Choice of ji.- TIn ref-rer es ", and )., the .a'ie
of t he >cr ndary -l-arer shI a s.n-r te-r was r estric 'te
to 1.4. If ec-iuati:n (1J4 is uJst, is i'en a
91
arbitrar.-. increa-ent Hi, and 0 tren the chance
in r.a'Y be ei'Ten b-y


kr.
\L Y


T_ ,



where T. !ies bc.-tveen the :rr'xim~r and rrinlmt.um velocity

in the inter-al ::, :x. The term i-] has been

Cc;F rID TnTII.AL







N.ACA ACR :D. Lo-422


eval.:uated for incompressible flow and a straight-line
velocltg distribution

U = Tx0 bx


The expression for


becomes


Si -Al"[,(Hi+l)(n+l)+2+ i (n+1l)




(!-n+2
U--
II '1 It j ^ri~i ~ tn+l)( + + l 1

ii I

(i + 1)(n + 1) + 2
(17)
(i + 1)(n + 1) + 2 + AHj(n + 1)

Although equation 17) is for ircompressible flow and the
exoression for -- x is only a: roximstely correct for

cases in w-ic. -- 0, equation (17) indicates that
TE


excessive errors in 8/L are possible if a in equation (9)
is made to equal zero and Hi is replaced by Fc andrestricted
to a constant cv--sseed value when the actual variation of
Hc is large. Irnrea.sin the body thickness, the lift
coefficient, or the M~rach number increases the variation of
Hc over the :.iface.

In the integrated equations the value of He may
vary over the surface but that of H- is assumed to be
constant. Because Hi can usually be estimated to withinn
0.2 for cases in -';Sich flow separation does not occur, the
error in /. that ?may be caused by a poor choice of Hi
is unlikel'- to be more than 10 percent.

-"hen the velocity increases in the direction of flow,
the value of Ei may be taken ss 1.2. When the velocity
decrcses in the direction of flow, the value of Hi
should b increased fro.n l., for cases in which thie total
change of velocity is aboii 20 p-rcent of the initial
velocity, to values near 1,71 for cases in which the total
change in velocity exceeds 0 percent of the initial
velocity. Because the velocity profile for the laminar


COC "IDEHTIAL


CONFIDENTIAL








N.CA A.Cx No. L6A22 CrnFIrT:ITIAL, 23


bourndi .ry 2. ~l;r -has been res tr. te.d tc th"e ilasius flat-
plate profile, Fi is equal to 2. Fo,2 the larri.nai-
b..iu.-ry lawyer, a then has Yte value i..

Full thickl.ness3 of brirndary la"-r.- Thse 'ui 1 thickne s
of r-.-e t...rbulA r.t boE'nd r: ." l1-ay r in tch :ase of" t'.,o-
die-r.sioral fle''-r mr;- te obtainec y. .'se of the relation




For th.e hcd 4 f re'rc.l1t.ic r the, f'.ill thickness of the
turbu.lenu L o.:id r-.; _ay r ny a tsine., after
rtmax
is comT;_:,lted, r.. i.use cf c:Fe rel stion

1 + I -+ 6 2 cLs C


trm 2j- ) cos a.
\- C ((18)

Curves of _/6s and 62 aSginst K are given -n
figures 5 and 6 for four vsli .s of H .

The va:iattin of 9/6 for the PBslsi-s flat-nlate
profil- is ,i r.n in fi,'ura '7 for use i- ccm:u:ting the full
thickness of the lavina r- bo:L"..-r layer. For flow abcut
body of rev.oljticn, C/62 ay or n. lec r., since the
lam inar bo'unler:/ l19er's are usJally th-in. The expression
for- th full tLhick: ss of the la iTnar ::und.sr:y layer on a
booy of revolution thoreifo-e becomes



P f r \( rtj. \2 -t cf-


Per fi':. ,.:ve r the inner sr._:face f s rond channel
when the .b, und.?-:, 1- 7- v is thin, eauations (1K) and (1)
are rer-?.c e by


v () -- x ) cos a

rat -tm ax tCmax (2.

r2. : cos a(20)


C'T, IDENIT ,AL







. C'TID':TTAL NA1CA ACR No. L6o'


and


rt6 = (/ Nj (21)
r max tmax ( \ 21)

In order to obtain a general statement regarding the
effect of density changes on the momentum thickness, com-
outations of 8 were made for a linear velocity distri-
bution and a range of Mach number. The velocity distri-*
bution for 0 x ( 1 was defined by
-r-
U x
S- I1 + b-
Uo L

The only variable was Mo. The curves of 9 against Mo
are given in figure 8 for b = 0 and 0.2. The results
indicate that the effect of density variation becomes
important only at Mach numbers exceeding unity.


CONCLTTSIONS


A comparatively rapid method is presented for the
computation of boundary-layer momentum thicknesses for
flow over two-dimensional bodies, over bodies of revolu-
tion at zero angle of attack, and over the inner surface
of round channels all in compressible flow.

The computations indicate that the effect of density
changes on the momentum thickness in flows with pressure
gradients is small for subsonic flows.

Langley :'o;.:~iorial Aeronautical Lacoratory
National Advisory Committee lor aeronautics
Langley Field, Va.


C c;F TDTUITIAL









IHTACA ACR No. L6A22 CCIFT-riiITIAL 25





1. Yo'.ingp, D., and '"interbc ttomn, I. E.: Iote on the
Effect of C.ev;:re ssibilit' on the Profile Dras of
.ero..f'oill in Lhe Aibsence cf Shock "Ia'.'es. Re'.
'o. A. 5) R .A. ( Iiti.h) :.ay 1 110.

2. Pr:ndl, I.: The !'vechpics c.f iS--cous Fluids. 'cl. ITT
of kerc.dJ--nr mic' T'-ner' di'-. '~, sees. 1!. ;.nd 17.
.'.. '. D'ur.?nd, 6,<., Jl] i'-i.-? S'-.r,,/?F., ( Perl.in), 195" ,
pp. 8~ -:'lUIR anSd I C2-1 2.

Yun D. : LTh Calc:.ll:in of the Tc.tal and Skin
F'r-ition DragEs of Roies of Revolut'Ion at Zero
Iinci'-i. ,e. R7. .:. 3 '7i I'iti lh A. .C. 1959.

I. Squire, P. :... n Y .n .2. ..; The C sT l c..ia tion of
their of Aerrfoils. '. f- -i.A 1-i.,
L.r tish t.. ". ., 1 h'.

5. von Doen-hoff, Al'ert ., a<.- Tet-rvin, aeal: D
nation of G a-?'r ral el .aionrIs for tr: -.F Fe;i-i, ior c,f
'P ut ul rent E:;t.d'ry L ay7 '-. AI CT- ( :1 1 '31 19..

6. T'etervr.n, :Tes ]: A "eth-.od fior the Ranid Estir.iation of
Turlile .t So...ndarr-La-i er Thicl.es .es for Calci.lating
Prc.file Drr9;.. :.: ',. A,' 11 L lJC-14, 19 4 .

7. Fr'se]., .: ? cc'-. in Smooth Strai _ght Pine. at
Velccities s':ove ar cnd blow 3Sonc '";elocity. ITACA Ti
,u. 2 l S..

c. The -'ore-rn, Thecdore, a--id Re ier, K'thi.r-. Experiments
orn Dr a. cf -Rvolvir.- Dis-ks Cylner.s, 9nd Stream-
l.ie Reds It Hih Sceeds. ?IC, \CP l. [14_F 19,? .

9. Ieernar, Joseph F., ar .i e'.ran', 9 Fr: *t P.: Friction
in Pi :es at ES.-,,;:ler .i.. .c n:r .Si bts v 'tic Velocities.
liACA TV ,-. 9&.5, -'?'-

10. Fluic' titicr. Panel of the .o-ror,-..tiCa] Research
C ir t e a:;:.d= d Cthers: n "ern Die\lor.en ts in Fluid
Dr.na'n- i Vo, T., S. iold:rten, ed., The Clarendon
Tress (Cxf, rd,, r d'.


Tr.- -. -I L-rTT
-'- L i L -AT








C C::7I)TIAITT liNOA ACR No. L6A22


11. Brainerd, J. G., and -rmnons, H. 7.: Effect of Vari-
able Viscosity on Boundsry Layers, wi t. a i-
cussion of Drag Measurements. Jour. Appl. Mech.,
vol. 9, no. 1, March 1942, pp. A-i A-6.

12. Falkner, V. M.: A New Law for Calculating Drag. The
Resistance of a Smooth Flat Plate with Turbulent
Boundary Layer. Aircraft Fnlr.eering, vol. XV,
no. 169, March 1945, pp. 65-69.


CC' IT-'IT IAL







NACA ACR No. L6A22 Fig. 1








us

0'
OU

4


Z.


z 8



1 0
0


0


o.

Z UJ
w
W


o0 0
u zo





Y-4-
0







*H
r74






Fig. 2 NACA ACR No. L6A22




%a



w- .
II M. 0


S--"-------- -z
4 .




---- ------ --= a




oo



o
O z




e-- ---- -- ---- 4





z --


-t
\10






-\ \- 0 ,




0
^-------------------------------- ---< r-




















___h__-
V4








NACA ACR No. L6A22


Fig. 3



0





4-1
43
Cd








1-I
4-1







C
1'1

















H
43
ca









PQ




cd


0








1-4
1-1
:4
- 4o
d
0















N4







NACA ACR No. L6A22


'00
si

0
p
14
d

0



q-4




S-dt
Cn
SO





1k
N) u



N'


Fig. 4






NACA ACR No. L6A22


- II


0
A


a
ZZZiIiZii I--------z0

----iii -zn'
_- __ _


U-


9-



I.--
9-
K-
8-


--z
L.
-o


-- -

z
0---

-Pl-


I I I I I I I I


~- N ~
N N N N.




Fig. 5


.1I
Ir)


Iri
w
43

o




0


o




4J
ra
-4









43
0
p,






to
O










I*4







NACA ACR No. L6A22


-_ -- _---- a.
P"
0






U

















2-~ ---
So
04












I II







u:~~ -
I-.-


2 -- --- --- --


o L::!
I. -- -0.


cIq


Fig. 6


g
fr-


N





*"I



4,
CM




I
%H






0
o
-r4



rM4
I




h0
T-l








NACA ACR No. L6A22


U

-O
Sal



04




zo
Z-





















O
z
- - -ZLL








__.

-- U.


r
,-H










CH





U,
0
r4
0


(l












0





S
a,

0n
0












6-4
F4
r F





) <


bLa


Fig. 7






NACA ACR No. L6A22


.o oze




.oozo
.0068
.0024


.0020

.00/6


.00[2


.0006


.0004

0


CONFIDENTIAL




b
-0.2














CONFIDENTIAL


/o


NATIONAL ADVISORY
COMMITTEE FOR AERONAUTICS


Figure 8.- Variation of
button given by U =
Uo
constants in equation
RL = 107; w = 0.768;

= O.


B/L with Mo for velocity distri-


1 + bL; 0 < x/L <1.
L


Values of


(14): k = 0.006555; n = 1/6;
y = 1.4; Hi = 1.4; a = 0.78;


Fig. 8













UNVERSTY OF FLORIDA
DOCUMENTS DEPARTMENT
1 .-RSTON SCIENCE UBRPARy
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GAINESVILLE, FL 32611-7011 USA